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Classification of solvable groups possessing a unique nonlinear non-faithful irreducible character. (English) Zbl 1287.20013
Summary: In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give examples to show that these groups need not to be solvable in general.

MSC:
20C15 Ordinary representations and characters
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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[1] Aschbacher M., Finite Group Theory, Cambridge Stud. Adv. Math., 10, Cambridge University Press, Cambridge, 1986
[2] Iranmanesh A., Saeidi A., Finite groups with a unique nonlinear nonfaithful irreducible character, Arch. Math. (Brno), 2011, 47(2), 91-98 · Zbl 1249.20009
[3] Seitz G.M., Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc., 1968, 19, 459-461 http://dx.doi.org/10.1090/S0002-9939-1968-0222160-X · Zbl 0244.20010
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